To improve the rate, additional assumptions on the measures \((,)\)  are necessary.
The rate can be refined by replacing the dimension \(d\)  of the ambient space (e.g. \(=^d\) ) by the intrinsic dimension (such as the Hausdorff dimension) of the input measures [1]}, and by the lowest intrinsic dimension of the two considered inputs [2]}.
Considering densities smooth to the order \(p\)  reduces the complexity to \(O(N^{-p/d})\)  [3]}, but the estimators \((_N,_N)\)  rely on wavelet decomposition or kernel estimation, which complicates the computation of \((_N,_N)\) .
Another approach consists in estimating \(\)  using an approximation of the optimization program itself.
Other approximations are possible such as an SDP relaxation which leverages the smoothness of inputs [4]}.
For unbalanced OT in [5]}, similar statistical complexities than OT can be obtained.
Note also that breaking the curse means a rate \(O(N^{-r})\)  where \(r\)  does not depend on \(d\) , but the constants might depend exponentially on \(d\) , as exhibited in [4]}, as well as for regularized OT (see next paragraph).
